The box gave him a purpose for continuing to live, so he could return it to its rightful owner. He had a hope that one day, he would get off the island, and eventually, when he realized no one was ever going to come for him, he got the strength to head out, against all odds. Otherwise, his life on the island was relatively safe, he had figured out how to survive, but was alone and very lonely. Not opening the box was like a last holdout on his morals, because otherwise, on an island where nothing matters, our morals can degenerate. In the movie, the lead character did not display any religious faith at all. He never prayed, he never asked God for help, he never thanked God for saving his life. So all he had was Wilson for a friend, and the box that belonged to someone else.

I'm sure others have analyzed the movie and probably on a much more profound level, but that is just what I have thought about it.